Induced polarization as a tool to non-intrusively characterize embankment hydraulic properties

HAL Id: hal-03005847

https://hal.archives-ouvertes.fr/hal-03005847

Submitted on 14 Nov 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

characterize embankment hydraulic properties

A Soueid Ahmed, A Revil, F Abdulsamad, B Steck, C Vergniault, V Guihard

To cite this version:

A Soueid Ahmed, A Revil, F Abdulsamad, B Steck, C Vergniault, et al.. Induced polarization as a tool to non-intrusively characterize embankment hydraulic properties. Engineering Geology, Elsevier, 2020, 271, �10.1016/j.enggeo.2020.105604�. �hal-03005847�

Contents lists available atScienceDirect

Engineering Geology

journal homepage:www.elsevier.com/locate/enggeo

Induced polarization as a tool to non-intrusively characterize embankment hydraulic properties

A. Soueid Ahmed^a, A. Revil^a,⁎, F. Abdulsamad^a, B. Steck^b, C. Vergniault^c, V. Guihard^b

aUSMB, CNRS, EDYTEM, Université Grenoble Alpes, 73000 Chambéry, France

bEDF R&D, 6 quai Watier, 78400 Chatou, France

cEDF DI-TEGG, 905 avenue du Camp de Menthe, 13097 Aix-en-Provence, France

A R T I C L E I N F O Keywords:

Induced polarization Electrical properties Permeability Earth dam Petrophysics

A B S T R A C T

Characterizing key petrophysical parameters of dams and embankments (including water content, specific surface area or cation exchange exchange capacity, and permeability) is an important task in estimating their degree of safety. So far, induced polarization tomography has not been investigated to check if it can play such a role. We have conducted a time domain induced polarization profile along the embankment of a canal in the South East of France. The profile is 560mlong. It comprises 1696 apparent resistivity and chargeability data and was accomplished by separating the current and voltage electrode cables to improve the signal-to-noise ratio. In order to complement the study, we performed induced polarization measurements on six core samples (including a clayey material and five carbonate rocks) collected from outcrops. A petrophysical induced polarization model called the dynamic Stern layer model is tested to see how the electrical conductivity and the normalized chargeability can be both connected to the porosity, the cation exchange capacity (CEC), and permeability of these materials. Then, these results are applied to interpret the electrical conductivity and normalized charge- ability tomograms into water content and CEC tomograms. In turn, these two parameters are used to compute a permeability index.

1. Introduction

Earth dams and embankments play an important role in water re- sources management and flood remediation. These infrastructures need continuous maintenance and meticulous monitoring in order to detect as soon as possible any anomaly that may affect their integrity (e.g.,Fell et al., 1992, 2003;Lee et al., 2005). In this context, knowing the ma- terials property distribution of these infrastructures provides valuable information to monitor their safety. Geophysical investigations, espe- cially geoelectrical methods, offer the possibility of non-intrusively imaging of Earth dams and embankments (e.g.,Panthulu et al., 2001).

In this context, the self-potential (Rozycki et al., 2006;Soueid Ahmed et al., 2019), electrical resistivity (Fransisco et al., 2018;Himi et al., 2018) and mise-à-la masse methods (Ling et al., 2019) have played a great role in localizing leakages in dams and embankments.

Induced polarization is a non-intrusive sensing technique used to investigate the low-frequency (< 100 Hz) polarization mechanisms occurring in porous media after being exposed to a primary current or electrical field (e.g.,Lesmes and Frye, 2001). By polarization, we mean

here the reversible storage of electrical charges in porous media through mechanisms that are non-dielectric in nature (seeRevil, 2013a, 2013b). The induced polarization method was first described by Schlumberger (1920) but its use was, for a long time, restricted to mineral exploration for reasons related to the high signal-to-noise ratio of induced polarization for such applications (e.g.,Bleil, 1953;Pelton et al., 1978;Zonge and Wynn, 1975). In the last decades, induced po- larization has been increasingly used in the realm of environmental engineering because of the development of more sensitive equipments and better metrological developments to improve the signal-to-noise ratio of the method (e.g.,Börner et al., 1996;Slater and Lesmes, 2002;

Kemna et al., 2012;Weller et al., 2013). In addition, such developments have benefited from a better understanding of the physics of the un- derlying polarization mechanisms in porous materials (e.g., Rosen et al., 1993;Revil and Florsch, 2010;Revil, 2012;Revil, 2013a, 2013b).

Furthermore, induced polarization can be used to predict, to some ex- tents, the permeability of porous media (Scott and Barker, 2003;Weller et al., 2010) and may therefore be useful to characterize dams and embankments hydraulic properties.

https://doi.org/10.1016/j.enggeo.2020.105604

Received 1 January 2020; Received in revised form 19 March 2020; Accepted 20 March 2020

⁎Corresponding author.

E-mail addresses:abdellahi.soueid-ahmed@univ-smb.fr(A. Soueid Ahmed),andre.revil@univ-smb.fr(A. Revil),feras.abdulsamad@univ-smb.fr(F. Abdulsamad), barthelemy.steck@edf.fr(B. Steck),christophe.vergniault@edf.fr(C. Vergniault),vincent.guihard@edf.fr(V. Guihard).

Engineering Geology 271 (2020) 105604

Available online 24 March 2020

0013-7952/ © 2020 Elsevier B.V. All rights reserved.

T

In time domain induced polarization, a box (primary) current is injected in the ground for a periodTusing a pair of electrodes A and B (source and sink). The difference of electrical potential is measured using several bipoles of electrodes M and N. The secondary voltage on a bipole MN decays over time after the shut-down of the primary current.

The existence of such a decay reflects that the ground behaves as a capacitor, storing reversibly electrical charges. The strength of this polarization is characterized by a physical parameter called the char- geability. For the frequency-domain induced polarization (often called spectral induced polarization), a complex electrical conductivity is measured with an in-phase component describing pure conduction phenomena and an imaginary or quadrature component reflecting pure polarization phenomena. The quadrature conductivity and the nor- malized chargeability (the product of the chargeability by the electrical in-phase conductivity) are proportional to each other. This relationship can be used to connect frequency and time-domain induced polariza- tion (e.g., Revil et al., 2017a). In the field, time domain induced po- larization measurements are favored due to the swiftness of their ac- quisition compared to spectral induced polarization measurements. In the laboratory, we usually prefer using spectral induced polarization but the relationship between the two approaches are pretty well-es- tablished (e.g.,Kemna et al., 2012).

The main scientific question we want to tackle in the present study is to investigate the possibility of discriminating lithology and material properties (permeability, water content, cation exchange capacity) in earth dams and embankment structures using the induced polarization method. In this context, it is important to recall the limitation of the DC resistivity method alone, however, broadly used to characterize earth dams and embankments. Electrical conductivity depends on two con- tributions, a bulk conductivity (mostly dependent on the water content and salinity of the pore water) and a surface conductivity mostly con- trolled by the cation exchange capacity (CEC) of the material, which is proportional to its specific surface area. Electrical conductivity/re- sistivity cannot be used as a stand-alone technique to separate these two contributions. Induced polarization offers a remedy to this issue and can be used in concert with resistivity to separate the effect of the water content from the effect of the CEC when the salinity of the pore water is known or independently estimated.Abdulsamad et al. (2019)recently performed a first step in this direction. They worked with an experi- mental dam located at Aix-en-Provence (South of France). They used a controlled leakage area and showed that the induced polarization method is capable of imaging the changes in the water content asso- ciated with the leak.

In the present paper, we use the induced polarization method for characterizing the material properties of a portion of the embankment of a canal in the South-East France. We investigate to what extent the induced polarization method can be used to delineate the lithology and image different key petrophysical parameters of the embankment structure. To the best of our knowledge, this is the first attempt to infer such physical properties in such environments from induced polariza- tion tomography. Independent petrophysical measurements are per- formed using 6 samples collected on the site to see if induced polar- ization can distinguish between different lithologies and they are connected to porosity, CEC, and permeability. In turn, we develop a permeability index that we test in the field.

2. Petrophysical model and experiments 2.1. Theory

As mentioned above, induced polarization refers to the reversible storage of electrical charges at grain scales in a porous medium under the influence of the electrical field. Such low frequency polarization is associated with the existence of an electrical double layer coating the surface of the grains (e.g.Phan et al., 2004,Fig. 1). This double layer is itself composed of an external layer called the diffuse layer and an

internal layer called the Stern layer. We first summarize the results of a petrophysical polarization model called the dynamic Stern layer po- larization model (seeRevil, 2013b, for a theoretical development). We consider a metal-free porous material. We apply to this porous material an external (primary) harmonic electric fieldE=E0exp (+iωt),E0

(Vm⁻¹) denotes the amplitude,ωdenotes the pulsation frequency (in rad s⁻¹), andt(in s) is time. Under the action of this applied electrical field, the grains get polarized, which generates a secondary electrical field in the opposite direction of the primary electrical field. In this condition, the porous material can be characterized by a frequency- dependent complex conductivityσ^∗entering Ohm's law,

=

J E, (1)

whereJandEdenote the current density (A m⁻²) and the electrical field (V m⁻¹), respectively. The complex conductivity is given by (Revil et al., 2017a, 2017b)

= M +h

i d

( ) ( )

1 ( ) ,

n

0 1/2

(2) whereωis the angular frequency (rad s⁻¹),τis a relaxation time (in s), andh(τ) denotes a probability density for the distribution of the re- laxation times associated with charges accumulations at grain scales. In Eq. (2), the quantity M_n denotes the normalized chargeability (ex- pressed in S m⁻¹) and is defined byMn≡σ∞−σ0whereσ∞andσ0

(both in S m⁻¹) denote the instantaneous and DC (Direct Current) conductivity of the porous material, respectively. They also correspond to the high (typically above 100 Hz) and low (typically around 0.1 Hz) frequency value of the electrical conductivity. In time-domain induced polarization, the quantityσ∞corresponds to the conductivity just after the application of the external (primary) electrical field. In this situa- tion, all the charge carriers are mobile (Fig. 1andRevil et al., 2017a).

The quantityσ0(S m⁻¹) defines the conductivity of the material for a long application of the electrical field corresponding to steady-state conditions.

According to the theoretical model developed byRevil (2013b)we have,

=s + F

s

F BCEC,

wn

w wn

g 1

(3) Fig. 1.Polarization of a clay grain under the effect of an applied electrical field.

The polarization of the grain is responsible for a phase lag (between the elec- trical field and the current) in spectral induced polarization and a secondary voltage that can be observed after the (primary, applied) current is shut down in time-domain induced polarization.

=s + F

s

F (B )CEC.

wn

w wn

0 g

1

(4) In these equations,F(dimensionless) denotes the intrinsic formation factor related to the porosityϕ(dimensionless) by the first Archie's law F=ϕ^-m(generalized toF=aϕ^-mfor a collection of rock samples),m (dimensionless) is called the first Archie exponent or porosity exponent (Archie, 1942), n (dimensionless) is called the saturation exponent (dimensionless, generallym≈n),sw(dimensionless) denotes the water saturation related to the (volumetric) water content byθ=swϕ,σw(in S m⁻¹) is the pore water conductivity,ρgdenotes the grain density (in kg m⁻³, usuallyρg= 2650 kg m⁻³), and CEC (C kg⁻¹where C stands for Coulomb) denotes the cation exchange capacity of the material. This CEC corresponds to the density of exchangeable surface sites on the surface of the mineral grains. It is typically measured using titration experiments in which the surface of the grains is exchanged with a cation having a high affinity for the sites populating the mineral sur- face. It is often expressed in meq/100 g with 1 meq/100 g = 963.20C kg⁻¹. In Eqs.(3) and (4),B(in m²s⁻¹V⁻¹) denotes the apparent mo- bility of the counterions for surface conduction. By surface conduction, we mean the conductivity associated with conduction in the electrical double layer coating the surface of the grains. The quantity λ (in m²s⁻¹V⁻¹) denotes the apparent mobility of the counterions for the polarization. The surface conductivity corresponds to the last term of Eq.(3)and is written asσS. A dimensionless number R has been in- troduced byRevil et al. (2017a)R =λ/B. From our previous studies (e.g., Ghorbani et al., 2018), we have Β(Na⁺, 25 °C) = 3.1 ± 0.3 × 10⁻⁹ m²s⁻¹ V⁻¹ and λ(Na⁺, 25 °C) = 3.0 ± 0.7 × 10⁻¹⁰m²s⁻¹V⁻¹, andRis typically around 0.09 ± 0.01 (Ghorbani et al., 2018). Eqs.(3)–(4)together with the definition of the normalized chargeability leads to,

=

M s

F CEC.

n wn

g 1

(5) The chargeability itself is a dimensionless quantity defined by M=Mn/σ∞.

In order to bridge time-domain and frequency-domain induced po- larization, a quantitative relationship between the normalized charge- ability and the quadrature conductivity is required. Considering the quadrature conductivity as the geometric mean of two frequenciesf1

andf2and the normalized chargeability defined as the difference be- tween the in-phase conductivity at frequencyf2 >f1and the in-phase conductivity at the lower frequencyf₁, we have (Van Voorhis et al., 1973)

f f M f f

"( _{1 2}) ⁿ( , )^{1 2} , (6)

2 A

ln , (7)

whereAcorresponds to the number of decades separatingf1andf2. For instance for two decadesf2= 100f1, we haveA= 10²andα≈ 3. For 6 decades (which is typically the broader range of frequency of interest in induced polarization), we haveA= 10⁶andα≈ 9.

As discussed later in this paper, in the field we measure apparent resistivity and chargeability data on a set of electrodes. After being inverted, these provide tomographic images of the conductivityσ∞and normalized chargeabilityMndistributions of the subsurface. This means that for each cell used to discretize the subsurface, we have two mapped parameters. Based on the previous petrophysical model and assuming m=n(i.e., the cementation and saturation exponents are equal to each other, see justification inRevil, 2013a, 2013b, his fig. 18), we can de- termine the two unknowns of the problem, i.e. the water content and the cation exchange capacity (CEC) distributions (seeAppendix Afor details),

= M

R

1 ,

w

n 1/m

(8)

= M CEC _m ⁿ ,

1g

(9) whereθ= swϕ denotes the water content and wherem can be de- termined from petrophysical measurements as done in the next section (a value ofm= 2 can be taken by default). Sinceλ,R, andρ_gare all well-determined constants as discussed in the next section, Eqs.(8) and (9)require only the measurements ofσ∞andMnand the knowledge of m.

2.2. Laboratory experiments: procedures

The samples used in this study were collected at outcrops of the two formations visible at the ground surface of the test site discussed in Section 4below (Figs. 2 and 3a). Sample #1 corresponds to a clayey soil. Samples #2 to #6 correspond to carbonate rock samples from a carbonate outcrops in the first section of the geophysical profile. The (connected) porosity was estimated from the volumes of the core samples and their dry and water-saturated weights. Their cation ex- change capacity was estimated by the Hexamine cobalt chloride method and spectrophotometric titration (Ciesielski et al., 1997;Aran Fig. 2.Position of the geophysical profile along the embankment of the channel with the position of the carbonate outcrop and the clayey soil sample collected for the laboratory measurements.

et al., 2008). The complex conductivity experiments were made with the ZEL-SIP04-V02 impedance meter (Fig. 3b, see also Zimmermann et al., 2008). The spectra were obtained in the frequency range 0.01 Hz–45 kHz. They are reported inFig. 4. The conductivity mea- surements are performed at three salinities: the in situ pore water conductivity using the water from the canal and two higher salinities using NaCl brines.

To estimate the surface conductivityσSand the formation factorF, we write Eqs.(3) or (4)as

= +

F

1 _{w S}, (10)

In order to avoid to give too much weight to the high salinity conductivity data, we fit the data as X= log10(a10^Y + b) where X= log₁₀σ,Y= log₁₀σ_wand the two model parameters area= 1/F andb=σS. The model parameters are obtained with a standard least square minimization technique. The fit of the curves is shown inFig. 5.

The porosity, CEC, formation factor, surface conductivity, normalized chargeability and quadrature conductivity are reported inTable 1.

2.3. Results

InFig. 6, we plot the formation factor versus the porosity and we fit the trend with a generalized Archie's lawF=aϕ^−mwhereaandmare determined using a least square optimization procedure. InFig. 7, we plot the surface conductivity and the normalized chargeability as a function of the reduced CEC defined as CEC/Fϕ(see Eqs.(3) and (5)).

The fit of the data provides the value of the mobilitiesBandλ. The good quality of the linear trends validates the theoretical predictions.Fig. 8 validates Eq.(6). According to Eq.(7), the coefficient α should be equal to 9, which is fairly close to the coefficient determined in Fig. 8 (α = 12 ± 4).

2.4. Permeability

The knowledge of the permeability distribution of an embankment is an essential ingredient to determine the potential for seepages.Sen et al. (1990)proposed the following relationship to determine the value of the permeabilityk(expressed in m²) from the cation exchange ca- pacity CEC (in C kg-1) and the porosityϕ(dimensionless) thanks to the following semi-empirical equation,

=

k k (1/FQV) ,c

0 (11)

where k0 and c are two fitting parameters, F ≈ ϕ² and QV = ρg

(1 −ϕ)CEC/ϕ withρgthe grain density close to 2650 kg m⁻³. The linear fit (least square regression) marked by the solid line inFig. 9 yieldsk≈ 10^4.30(1/FQV)². Accounting for the dependence of the ap- parent formation factor and excess of charge per unit pore volume with the water content, Eq.(11) can easily be generalized to unsaturated conditions to

k k

( CEC) ,

g 0 6

2 (12)

withk0= 10^4.30andkis expressed in m². From Eq.(12), it is clear that permeability in the vadose zone is extremely sensitive to the water contentθas expected. Along an embankment or in a dam, leakages would take permeable pathways that would be fully water-saturated.

Therefore, we can determine a permeability index as

I k

k Max(log )

log ¹⁰ ,

10 (13)

We will test this formula for the case study explored in the next section of this study.

3. Tomography: modeling procedure

In this section, we present and discuss time-domain induced polar- ization (TDIP) measurements and data inversion. This procedure will be applied to the field data inSection 4.

3.1. Macroscopic equations

TDIP measurements are favored in the field because they require less acquisition time and the current technology of resistivity meters allows induced polarization measurements to be performed simulta- neously with resistivity measurements (Fig. 10). A TDIP experiment consists in injecting a currentI(t) (in A,tis time) between two elec- trodes A and B an then recording the voltage decayδψMN(t) between two potential electrodes M and N. Usually we prefer having the AB electrodes and the MN electrodes on two distinct cables to avoid elec- tromagnetic (capacitive and inductive) coupling effects (Fig. 10). Once the primary current is shut down (for t > 0), only the secondary Fig. 3.Carbonate core samples used in this work from the test site and impedance meter used for the spectral induced polarization measurements. (a) We have 6 samples. The first sample (not shown here) is a clayey soil, which is unconsolidated. The five other samples are carbonate-rich rocks collected in the field at an outcrop at the beginning of the induced polarization profile. (b) Impedance meter ZEL/SIP04-V02 (Zimmermann et al., 2008).

voltage persists. It is decaying over time while the charge carriers are coming back to their equilibrium position. This decaying secondary voltage is measured into windows (W1, W2, etc.) separated by char- acteristic times (t0,t1,t2, …). The partial chargeabilities are determined for each of these windows by integrating the secondary voltage over time. T_onand T_offdenote the time of current injection and time for potential decay measurement, respectively. The timet0is a delay (dead) time before starting the partial chargeability measurement.

The response of a 3D conductive medium to a given current ex- citation can be described by the following elliptic partial differential equation:

=

x y z I x x y y z z

( ( , , ) ) ( s)( s)( s), (14)

whereσ(x,y,z)(in S m⁻¹) denotes the electrical conductivity distribu- tion,ψ(inV) denotes the electrical potential generated by the injection of the current,δis the Dirac distribution (used to simulate the injection of current at punctual sources or sinks A and B),xs,ys, andzsare the spatial coordinates of the current sources or sink A and B. To solve Eq.

(14), we need to specify the boundary conditions. We generally use the

following boundary conditions:

=0on ,_D (15)

=

n 0on N, (16)

whereΓ_DandΓ_Nare known the Dirichlet's and Neumann's boundaries, respectively,Γ_D∪Γ_N=Γ(Γrepresents all the boundaries of the do- main),nis the outward unit vector that is normal to the boundaryΓ_N. Physically, Eq.(16) means that when we get away from the area of current sources and sinks, the potential should drop to zero, while Eq.

(16) refers to electric insulation and is applied at the top of our domain to simulate the ground/air interface. In the present paper, we will work on 2.5D induced polarization profiles. In this case, we still need to account for 3D effects and this could be achieved by simulating an in- finite extension to the strike direction (i.e., the direction normal to the profile). We refer to this space as the 2.5 dimension. An efficient manner of solving Eq. (14) in 2.5D is obtained by solving it in the Fourier domain (e.g.,Dey and Morrison, 1979):

+ =

x z x y z k x z x y z I

x x z z

[ ( , ) ( , , )] ( , ) ( , , )

2 ( s) ( s)

2

(17) where is the Fourier transform ofψandkis the wavenumber. The potential is then obtained by:

=

x y z x k z ky dk

( , , ) 2 ~( , , )cos( ) ,

0 (18)

which numerically is computed through:

= ( )k

n

n n

(19) whereωnare weights that can be computed following the method re- ported in (Press et al., 1992). The macroscopic mechanisms of the TDIP problem can be described though the introduction of an intrinsic electrical property known as the chargeabilityM (i.e.,Seigel, 1959).

This parameter indicates the capacity of the medium of reversibly 10^-6

10^-5 10^-4 10^-3 10^-2

10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10^-3

10^-2 10^-1 10⁰

10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 1

2 3 4 5 6

Frequency (Hertz)

Quadrature conductiivty (S/m)In-phase conductiivty (S/m) Samples a.

b.

Fig. 4.Complex conductivity spectra of the 6 core samples considered in this work. (a) In-phase electrical conductivity spectra. (b) Quadrature conductivity spectra. The presence of several peaks can be associated to the existence of different peaks in the pore size distribution.

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

-1.5 -1 -0.5 0 0.5 1 1.5

1 2 3 4 6

In situ

Samples

Log(pore water conductivity, S/m)

Log(rock conductivity, S/m)

Clay

Carbonates

Fig. 5.Rock conductivity (at 1 Hz) versus pore water conductivity for some of the core samples used in this study. In situ represents the value of the in situ pore water conductivity in the canal along the embankment studied in the present paper. From these conductivity curves, we determine the formation factor and the surface conductivity in a log-log space using a non-linear fitting procedure based on a least-square technique. We also observe that at the in situ water conductivity, there is a strong effect of the surface conductivity.

storing electrical charges after been subject to a primary current.Seigel (1959)proposed to describe the polarization effects as a perturbation of the electrical conductivity. We can therefore compute a potential dis- tribution that accounts for the polarization effects defined as:

=G( (1₀ M)) (20)

whereGis the operator associated to Eq.(14)in 3D and to Eq.(14)in 2.5D.

As reported inOldenburg and Li (1994), the apparent chargeability can be computed as:

= Ma 0

(21) Withψ₀=G(σ0). Eq.(21)means that in order to compute the in- duced polarization response we only need to solve Eq.(17)twice, one with conductivity of them medium and one using the conductivity that is perturbed by the chargeability. The apparent (partial) chargeability can be computed as a function of the decaying voltage generated after the current shutdown as:

= ⁺

M 1 t dt

a ( )

t t

0 _i MN

1

(22) where δψ_MN(t) is the time dependent decaying voltage recorded be- tween electrodes M and N.

Table 1

Material properties of the rock samples.

Sample Porosityϕ(−) CEC (meq/100 g) F(−) σS(S m⁻¹) Mn(S m⁻¹) σ" (S m⁻¹)

1 0.30 ± 0.03 9.00 2.9 0.11 0.010163 165.0 × 10⁻⁵

2 0.20 ± 0.03 0.97 16 0.0038 0.00032331 3.20 × 10⁻⁵

3 0.20 ± 0.03 2.69 26 0.0080 0.0017825 7.86 × 10⁻⁵

4 0.21 ± 0.03 3.58 23 0.017 0.0020288 11.4 × 10⁻⁵

5 0.14 ± 0.03 0.32 63 0.00029 0.00016029 0.88 × 10⁻⁵

6 0.17 ± 0.03 0.10 34 0.00060 0.00024974 1.25 × 10⁻⁵

CEC denotes the cation exchange capacity (1 meq/100 g = 963.20C kg⁻¹),Fdenotes the formation factor (dimensionless),σSdenotes the surface conductivity (S m⁻¹),Mn(S m⁻¹) the normalized chargeability (with the DC conditions taken at a low frequency of 0.01 Hz and the instantaneous conductivity taken at the high frequency of 10 kHz), andϕthe connected porosity (dimensionless). The formation factor and the surface conductivity are determined using the rock conductivity versus the pore water conductivity in log log space. The quadrature conductivity is here provided at 10 Hz.

10⁰ 10¹ 10²

10^-1 10⁰

Formation factor (-)

Porosity (-) Clays Carbonates

0.25 2.8

F = φ^-

Fig. 6.Formation factor (dimensionless) versus (connected) porosity (di- mensionless) and fit with an empirical Archie's lawF=aϕ^−mto determine the values ofaandm.

10^-4 10^-3 10^-2 10^-1 10⁰

10¹ 10² 10³ 10⁴

Data Linear fit

=4.3 10 m s V´ ^{9 2 -1 1}

B ^{- -}

CEC / Fφ (C/kg)

Surface conductivity (S/m)

Clay

Carbonate

10^-4 10^-3 10^-2 10^-1

10¹ 10² 10³ 10⁴

CEC / Fφ (C/kg)

Normalized chargeability (S/m)

Data Linear fit

=0. 7 10 m s V´ ^{9 2 -1 1}

λ ^{- -}

Carbonate

Clay a.

b.

Fig. 7.Normalized chargeability versus CEC. (a) Surface conductivity versus the CEC divided by the tortuosity of the pore space (determined as the product of the formation factor by the porosity). The linear fit is done withσS=aCEC/

Fϕwitha= 13 ± 1 × 10⁻⁶in SI units andr= 0.98. (b) Normalized char- geability versus CEC/Fϕ(r= 0.97). The fits are used to determine the value of the mobilitiesBand λ.

3.2. Tomography

The tomography of the subsurface here refers to the process through which we seek to recover the physical parameters that give rise to the data that we measure. This is an inverse problem in which the model

vector corresponds to the physical parameters of interest (conductivity and chargeability) discretized over a grid representing a discretization of the subsurface in cells. In our case, the relationship between the data (i.e., the electric potential) and the unknown (i.e., the conductivity and the chargeability) is given by the nonlinear forward operator described in Eq.(14). The idea is to obtain the most optimal parameter set whose computed response matches the observed data and also does exhibit the heterogeneities of the subsurface. In practice, the inverse problem is discretized i.e., the domain of simulation is divided into cells, to each of which we assign a constant value of the physical parameter and the grouping of each these cells gives the global image of the field of the parameter in question. This means that we are not actually dealing with one unknown but with a set of unknowns that may contain several thousands of unknowns or several millions in case of large-scale ap- plications. The difficulty that arises within this context is to obtain a parameter model that has a large number of elements while the amount of data is generally limited and far smaller than the number of un- knowns. A common approach employed in the realm of geophysical inversion is to add information that may complete the one provided by the data and also copes with the possible lack of data and the noise it may contain. This strategy is known as regularization (e.g.,Tikhonov, 1943). Thus, the inverse problem could be casted into a minimization problem in which, we seek to minimize a misfit function given by:

= W d_d( ^obs d^sim) ²+ W m_m( m_ref) ² (23) whereWdis a diagonal covariance matrix accounting for errors in the data,d^obsis the vector of observed data,d^simis the vector of simulated data,Wmis the constraints matrix,αis a positive regularization para- meter that serves as a balance or weighting factor for giving different weight to the constraints term or to the data term,mis the discretized physical parameter vector,mrefis the initial a parameter set. The con- straints that we impose through the matrixWmdepend on our initial knowledge about the medium. If we are working on gradually variable media without abrupt change in physical properties, smoothing con- straints are favored. If we are in the presence of media with dis- continuities such as faults or factures, structural constraints are 10^-4

10^-3 10^-2 10^-1

10^-6 10^-5 10^-4 10^-3 10^-2

Normalized chargeability (0.01 Hz-10 kHz)

Quadrature conductivity (10 Hz)

M = 12 n σ”

Fig. 8.Normalized chargeability versus quadrature conductivity. In this plot the normalized chargeability is determined between a very low frequency (0.01 Hz) and a high frequency (10 kHz). The quadrature conductivity is de- termined at the geometric mean frequency of 10 Hz (see Eqs.(6) and (7)). The red plain line corresponds to the best fit used to determine the value ofα= 12.

(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

-18 -17 -16 -15 -14 -13 -12 -11

-11 -10 -9.0 -8.0 -7.0 -6.0

Basalts Sandstones

]m,ytilibaemreP[goL ^{Linear fit}2

Log 1 /éë FQ_V, C m^-3ùû

r² = 0.74

Volcanic rocks

Fig. 9.Permeability as a function of the formation factorFand the volumetric charge densityQV, which is related to the porosity and the cation exchange capacity. Data from:Niu et al. (2016),Revil et al. (2017a, 2017b),Ghorbani et al. (2018), andHeap et al. (2017), andSen et al. (1990). The linear fit (least square regression) yieldsk= 10^4.30(1/FQV)^2.09(r²= 074).

t_{, Time} Voltage between M and N

Secondary voltage decay

Ton Toff

Current:

yMN

y_¥ y0

Current I(t)

y0

-T 0

y(t)

t^0, t^1, t²

W1 W2

-T 0 t_{, Time}

A B

M N

Resistivity meter b.Spatial configuration

a.Time series in induced polarization

32 1

33 Current cable 64

Voltage cable

Fig. 10.Time-domain induced polarization data. (a) We inject the current be- tween electrodes A and B (a box signal of periodT). The potential difference between the electrodes M and N is the sum of a primary (instantaneous) voltage ψ∞and a secondary voltageψ0–ψ∞for whichψ0denotes the steady-state po- tential. (b) Spatial configuration for a time-domain induced polarization survey.

We used two sets of 32 electrodes for the current electrodes and 32 electrodes for the voltage electrodes. These two sets are located on two distinct cables to minimize spurious electromagnetic effects and electrode polarization effects.

privileged (e.g.,Gallardo and Meju, 2004;Zhou et al., 2014). An effi- cient and simple way of implementing the smoothing constraints is by choosing the matrixWmas a combination of the derivative operators that act as smoothing operators on each cell of the inversion mesh. As the inverse problem is nonlinear, it is iteratively solved and during each iteration, the model parameters are updated. In the present paper, we use the popular software Res2Dinv (Loke, 2002) for performing the 2.5 D TDIP inversion of the electrical conductivity and chargeability fields.

For more detailed information regarding the inversion process, the reader is invited to refer to (Loke and Barker, 1996a, 1996b;Loke and Dahlin, 2002).

4. Case study

The investigated canal is located in Provence-Alpes-Côte d'Azur region in southern France. The canal is classified A, which means that it is subject to a full technical exam each 10 years. Induced polarization measurements have never been operated on this embankment. The crest of the canal is 4mwide (Fig. 2). Its right bank is built on alluviums of different types including silt, sand, gravel and chipped stones. The left bank is mainly built on materials including limestones, marls, and argilites. This said, due to the topography, the structure of the canal is mainly composed of rockfills on its right bank and is composed of ex- cavation materials on its left bank.

The TDIP measurements were acquired on a portion of the left bank of the canal using a 4-channels ABEM SAS-4000 in July 2019 (Fig. 2).

We used two cables as shown inFig. 10. We performed a 560mlong induced polarization profile located at the middle of the crest of the left bank of the canal. This profile was acquired using the roll-along tech- nique which allows for setting up elongated resistivity or IP profiles.

Eight IP profiles, each of which is composed of 64 electrodes were concatenated to obtain the long profile (512 electrodes in total were used and the electrodes were moved by set of 16 electrodes to minimize the gaps in the pseudo-section, Fig. 2). The electrodes are separated from each other by a distance of 4mand the injected current amplitude is automatically adapted by the equipment but its minimal amplitude was 20 mA. The voltage decay is recorded on ten 100 ms time windows and after a dead time of 100 ms after cutting off the current to avoid capacitive and inductive effects. The period of the current injection is 1 s.A total of 1696 apparent resistivities and chargeabilities were ac- quired. For the acquisition of the induced polarization data, we fol- lowed the strategy proposed byDahlin et al. (2002)for improving the quality of the induced polarization measurements. This technique consists in using for each profile two parallel distinct cables, which means that injection electrodes are laid out along one cable and the measuring electrodes are laid out along the second cable parallel to the other one. The distance between the two cables is 1 m. Although this strategy reduces the depth of investigation, it however has the ad- vantage of avoiding the recording of the noisy current that may corrupt the voltages measured on the same electrodes that have been used both for injecting the current and measuring the voltage after the current has been shut down.Fig. 11shows an example of the decay curves that we have obtained on the field using this technique. The decaying shape of the curve gives an idea on the good quality of the acquired induced polarization data.

The simulation domain is discretized into 3542 rectangular cells, within each of them the physical parameters of the subsurface are es- timated. We used the commercial software Res2DINV (e.g.,Loke, 2004) for performing the inversion. We used a smoothing-constrained inver- sion based on the L2 norm. The inversions of the conductivity and chargeability fields are launched using the means of the apparent re- sistivity and apparent chargeability as starting models. The inversion process first estimates the conductivity distribution, and then it uses it

in the computation of the intrinsic chargeability distribution. The normalized chargeability tomogram is obtained by multiplying the conductivity by the chargeability on each cell of the inversion mesh.

Convergence was reached after 5 iterations.Fig. 12a shows the esti- mated conductivity field which reflects a shallow more resistive zone corresponding to the unsaturated area and deeper more conductive anomalies that represent the fully saturated area.Fig. 12b shows the normalized chargeability field. Higher normalized chargeability anomalies are visible in the deeper areas of the tomograms which co- incide with the presence of clayey materials. The shallow less charge- able area corresponds to the carbonates formation. To assess the quality of the inversion results, we report the root means square error (RMSE) for both the conductivity and chargeability fields. The conductivity RMSE is 16%, while the chargeability RSME is 6%. The high RMS error on the conductivity is due to the high input impedance of the ground when the measurements were done in summer 2019 during a heat wave. We could have smooth out the apparent resistivity to reduce the RMS error but we know by experience that such procedure does not change the final result/tomogram. The computed apparent resistivity and chargeability data reproduce the observed data with a good accu- racy.

InFig. 13, we plot the normalized chargeability as a function of the conductivity for both the field and laboratory data. The complete da- taset exhibits a first-order linear trend between the two parameters.

Low values correspond to the carbonates and high values correspond to clay-rich media. The black plain line corresponds to the expected maximum relationship between the normalized chargeability and con- ductivity withR= 0.10 as discussed inSection 2. Note that the la- boratory measurements are performed at full saturation at the in situ pore water conductivity (0.0512 S m⁻¹at 25 °C). The field data are obtained in the vadose zone, so in unsaturated conditions.

The CEC distribution (seeFig. 14b) is high in the clay-rich area with a maximum value close to 40 meq/100 g and a minimum value of 0.8 meq/100 g. This value is on the same order of magnitude than the measurement done in the laboratory for the clayey soil and close to 9 meq/100 g (Table 1). The water content distribution is shown in Fig. 14a. It indicates that the permeable formations are rather dry while the low-permeability formations are rather water-saturated likely Fig. 11.Apparent chargeability decay curves (expressed in mV/V) for few quadrupoles ABMN. We note the good quality of the measurements showing regular decay for the apparent chargeability with the elapsed time after the shutdown of the primary current.